Scale, distance and exact location

Grade 9 Social Studies Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.

2. 1.

   Two ports lie on 40°N. A ship sails due east along the 40°N parallel. Is that the shortest route, Esme asks?

   • No: the shortest route bulges north of the 40°N line toward the pole
   • No: the shortest route bulges south of the line toward the equator
   • Yes: staying on one latitude holds one heading, which is the shortest
   • Yes: each parallel is a great circle, so it is the shortest route
3. 2.

   Which comparison of Phoenix–Quito (5,130 km) and Phoenix–Charleston (2,780 km) is fair, Jun asks?

   • Quito is twice as far, since 5,130 is about double 2,780, a ratio of two to one
   • About 1.8 times; one sphere gives both, so the ratio is steadier than either
   • No ratio can be trusted, since Earth is not a perfect sphere and both distances are off
   • Quito is exactly 1.85 times as far, since the two figures are exact to the kilometer

4. Use this

   Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km. Card B. A flat-plane check. Using the city coordinates, a student gets Phoenix to Charleston as 2,794 km with a right triangle on a flat map. Coordinates are rounded to the minute, and one minute of latitude is about 1.85 km.

5. 3.

   A flat-plane method gave 2,794 km, and the sphere gave 2,780 km. What does the 14 km gap show, Kira asks?

   • The two must agree exactly, so one of them has an error
   • A flat triangle is close here: 14 km is about half a percent
   • The gap shows Earth is flat over distances of 2,800 km
   • The sphere figure is wrong by 14 km, so the flat figure is right
6. 4.

   A report gives Phoenix to Charleston as 2,780.4 km. What is wrong with the extra decimal, Luz asks?

   • Each end could shift about 1 km, so “about 2,780 km” is honest
   • The distance cannot be given, since rounding spoils the whole result
   • Nothing: 2,780.4 km is just 2,780 km with the decimal for accuracy
   • The decimal should be kilometers, 2,780 km and 400 m

7. Use this

   Card C. East–west distance. One degree of longitude spans about 111 km times the cosine of the latitude. The cosine of 60° is 0.50. Two towns lie on the 60°N parallel, 10° of longitude apart.

8. 5.

   Two towns on the 60°N parallel are 10° of longitude apart. How far apart are they along the parallel, Milo asks?

   • About 555 km: 10° at 111 × 0.50 km a degree
   • About 55 km: 10° at 5.5 km a degree
   • About 1,110 km along the parallel, and 555 km in a straight line
   • About 1,110 km: 10° at 111 km a degree

9. Use this

   Card C. East–west distance. One degree of longitude spans about 111 km times the cosine of the latitude. The cosine of 60° is 0.50. Two towns lie on the 60°N parallel, 10° of longitude apart. Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.

10. 6.

    For those two towns, how does the great-circle distance compare with the 555 km along the parallel, Noah asks?

    • A little shorter: the great circle bulges toward the pole
    • It is longer, since the great circle bulges and adds distance
    • It is the same, since both stay on the 60°N line
    • It is half as long, since the pole shortens the path

11. Use this

    Card D. Drawing roads. On a 1:1,000,000 map, 1 mm stands for 1 km. A country road about 10 m wide is drawn as a line at least 0.3 mm wide.

12. 7.

    Why is a 10 m road drawn 0.3 mm wide on a 1:1,000,000 map, Pia asks?

    • At true scale it would be 1 mm, so drawing it 0.3 mm shrinks it
    • The road is really 300 m wide, which is what 0.3 mm stands for
    • Widening a line makes the map’s scale wrong everywhere on it
    • At true scale it would be 0.01 mm, too thin to see, so it is widened
13. 8.

    Which measurement does drawing roads at 0.3 mm distort most on that map, Ren asks?

    • A road’s width; the distance between two town centers stays reliable
    • The map’s ratio scale, since line widths are part of a ratio
    • The length of the road, since a thick line is longer than a thin one
    • The distance between two town centers, since a wide line moves the centers

14. Use this

    Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.

15. 9.

    Tala’s note says: “On a Mercator map the shortest route is always a straight line.” Which fix is right?

    • Change it: the shortest route is straight, and a constant heading curves
    • Change it: the shortest route is straight on a globe, so it is straight on a map
    • Change it: a constant compass heading is straight; the shortest route usually curves
    • Leave it: Mercator is accurate, so a straight line is the shortest

16. Use this

    Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km. Card B. A flat-plane check. Using the city coordinates, a student gets Phoenix to Charleston as 2,794 km with a right triangle on a flat map. Coordinates are rounded to the minute, and one minute of latitude is about 1.85 km.

17. 10.

    Zee’s note says: “Phoenix to Quito is 5,130 km, so Quito is 5,130 km south of Phoenix.” Which fix is right?

    • Leave it: Quito is directly south, so the two distances match
    • Change it: Quito is 5,130 km east of Phoenix, not south
    • Change it: 5,130 km is direct; the south part is about 3,740 km
    • Change it: the south part is 5,130 + 544 km, adding Charleston’s gap